could any one tell me why not all automorphisms of a root system are elements of weyl group? For example in $A_n, n>2$ the automorphism $\alpha\mapsto -\alpha$ is not in the weyl group. I do not understand why? it is also a reflection and must be an element of the weyl group. thank you. well $A_n$ means root systm of rank $n$.
2026-03-27 13:25:50.1774617950
not all automorphisms of a root system are elements of weyl group
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The automorphism group of the root systems $Aut \Phi$ is the semi-direct sum of the Weyl group with the group of automorphisms of the Dynkin diagram. In the case of $A_n$ there exists a non-trivial automorphism of the diagram which flips over the two tails. It is not the one which sends $\alpha$ to $-\alpha$ as you claim. In the case of $D_4$ this group is isomorphic to $S_3$.